INTAS Final Report

Title

Constraint Scheduling� Mathematical models� algorithms� and ap�

plications

Reference number� INTAS������� ext

Project co�ordinator� Peter Brucker

Period� June ��� to May � ���

Objective of the Project

The objective of the project is to study classes of scheduling problems arisingin connection with production planning and with the design of computeroperating systems Such problems are more complex than classical schedulingenvironments because in most cases several additional constraints have tobe satis�ed In this project models representing these problems will beformulated and basic solution methods will be investigated

Research Activities

Description of the research activities

Most of the work done can be considered as basic research on schedulingproblems The objective was to formulate optimization models for solv�

ing complex scheduling problems in production planning and design of com�puter operating systems Results have been obtained for batching problemsscheduling problems for hybrid or �exible shops problems with operationsof equal duration or with a �xed number of jobs

Problems have been classi�ed according to their complexity For specialsituations polynomial algorithms have been developed For NP �hard prob�lems dynamic programming algorithms polynomial approximation schemesbranch�and�bound methods decomposition approaches and heuristics suchas local search methods have been applied Also stability analysis of optimalor approximate solutions have been performed

The research activities were carried out in accordance with those indicatedin the Work Program of the INTAS co�operation agreement

Results

Research was undertaken in seven areas The results obtained in each areaare documented in this section Workshop and conference activities are de�scribed Also a list of papers which have been prepared in the framework ofthe project and published or accepted for publication in di�erent journals isgiven

Area �� Batch scheduling

Batching or grouping of similar tasks is mainly done to avoid set�up times orset�up costs It is widely used in �exible manufacturing We refer to batchscheduling problems as those combining scheduling and batching decisions

A review of the literature on the models that integrate scheduling with batch�ing decisions is given in ��� Details of the basic algorithms are given andother signi�cant results are referenced Special attention is paid to the designof e�cient dynamic programming algorithms

A dynamic programming algorithm for the problem of scheduling jobs of twopart types in a shop consisting of three machines is presented in ��� Theprocessing of each job consists of two stages The �rst stage is undertakenon the machine common to all jobs and the second stage is undertaken on themachine speci�c to a particular part type Set�up times are necessary at the�rst stage to switch from processing a job of one part type to processing ajob of another part type Jobs of the same part type processed contiguouslyat the �rst stage form a batch The objective is to �nd a batch schedule min�imizing the makespan An integer programming formulation and a heuristicalgorithm was presented for this problem in the literature Our algorithmprovides an optimal solution in time polynomial in the number of jobs

The problem of scheduling n jobs on a single machine in batches to minimizesome regular criteria is studied in ��� Jobs within each batch are processed

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sequentially so that the processing time of a batch is equal to the sum ofthe processing times of the jobs contained in it Jobs in the same batch arecompleted at the same time when the last job of the batch has �nished itsprocessing A constant set�up time precedes the processing of each batch The number of jobs in each batch is bounded by some value b If b � n thenthe problem is called bounded Otherwise it is unbounded For both thebounded and unbounded problems dynamic programming algorithms arepresented for minimizing the maximum lateness the number of late jobs thetotal tardiness the total weighted completion time and the total weightedtardiness when all due dates are equal which are polynomial if there is a�xed number of distinct job due dates or processing times More e�cientalgorithms are derived for the unbounded problem of minimizing the max�imum lateness and some special cases of both the bounded and unboundedproblems in which all due dates and�or processing times are equal Severalspecial cases of the bounded problem are shown to be NP �hard Thus acomprehensive classi�cation of the computational complexities of the specialcases is provided

The problem of scheduling groups of jobs on unrelated parallel machinesin batches subject to group deadlines was considered in the literature Aclassi�cation of computational complexities of special cases was provided forthe situation when all groups have equal deadlines In ��� it is established thatthe problem is strongly NP �hard for the case of identical machines di�erentdeadlines unit processing times and unit set�up times and equal numbers ofjobs in each group It is shown that a presence of machine deadlines insteadof job deadlines does not make the problem easier

The single�machine problem which combines batch scheduling and a deter�mination of a common due date is studied in ��� An important special caseof this problem arises when there are equal set�up times and equal job pro�cessing times Computational complexity of this case was indicated to beopen however It is proved to be NP �hard

A new model with a so�called batching machine is considered in �� A batch�ing machine is a machine that can handle up to b jobs simultaneously Theproblem of scheduling n jobs on a batching machine to minimize regularscheduling criteria that is criteria which are non�decreasing in the job com�pletion times is studied The jobs that are processed together form a batchand all jobs in a batch start and �nish at the same time The processing timeof a batch is equal to the largest processing time of any job in the batch Two variants are analyzed� the unbounded model where b � n� and thebounded model where b � n For the unbounded model a characterizationof a class of optimal schedules is given which leads to a generic dynamic pro�gramming algorithm allowing the problems of minimizing various regular costfunctions to be solved e�ciently The unbounded problems of minimizing theweighted number of tardy jobs and the total weighted tardiness are proved tobe NP �hard For the bounded model an O�nb�b���� dynamic programmingalgorithm is derived for minimizing total completion time� and for the case

�

with m di�erent processing times a dynamic programming algorithm thatrequires O�b�m��m� time is given It is proved that due date�based criteriagive rise to NP �hard problems It is shown that an arbitrary regular costfunction can be minimized in polynomial time for a �xed number of batches

Di�erent constructive and iterative heuristic algorithms have been proposedin ��� for the permutation �ow shop problem with batch processing wherethe processing time of a batch consisting of jobs of the same group is givenby the maximum of the processing times of the operations contained in thebatch and the maximum batch size is limited As an objective functionboth the minimization of makespan and the weighted sum of completiontime has been considered Structural properties based on the critical pathhave been applied to guide the search in the case of makespan minimization The algorithms developed have been tested on problems with up to �� jobs

Multi�operation jobs are to be scheduling on a single machine are discussedin ��� Each job comprises several operations that belong to di�erent fami�lies Changeovers of production from one family to another have associatedset�up times A job completes when all of its operations have been pro�cessed This problem combines the batching aspect of the well�known familyscheduling models with an assembly element �where the job�s operations areassembled to produce the �nal product� Three classic optimality criteria�the maximum lateness the weighted number of late jobs and the sum of jobcompletion times are analyzed The problem of minimizing the maximumlateness is equivalent to its counterpart without assembly Thus complex�ity results and algorithmic solutions are deduced The number of late jobsproblem is binary NP �hard when there are two families and unary NP �hardwhen there are an arbitrary number of families even when all set�up timesare identical For a �xed number of families there is a a dynamic program�ming algorithm for the weighted number of late jobs problem which requirespseudopolynomial running time A similar algorithm solves the sum of com�pletion times problem to optimality under the additional assumption thatprocessing times are agreeable

References

�� Brucker P Gladky A Hoogeveen H Kovalyov M Y Potts C Taut�enhahn T van de Velde S Scheduling a batching machine Journal ofScheduling ��� Vol p ����

��� Cheng T C E Kovalyov M Y An exact algorithm for batching andscheduling two part types in a mixed shop� a technical note Interna�tional Journal of Production Economics accepted

��� Cheng T C E Kovalyov M Y Single machine batch scheduling withsequential job processing Working Paper The Hong Kong Polytechnic

�

University Faculty of Business and Information Systems No �������submitted to Journal of Scheduling�

��� Danneberg D Tautenhahn T Werner F A Comparison of Heuristic Al�gorithms for Flow Shop Scheduling with Setup Times and Limited BatchSize Otto�von�Guericke�Universit�at FMA Preprint �� ��� �submit�ted to EJOR�

��� Gerodimos A E Glass C A Potts C N Tautenhahn T SchedulingMulti�Operation Jobs on a Single Machine Annals of Operations Re�search submitted

��� Kovalyov M Y Batch scheduling and common due date assignmentproblem� an NP �hard case Discrete Applied Mathematics submitted

��� Kovalyov M Y Shafransky Y M Batch scheduling with deadlines onparallel machines� An NP �hard case Inform Proc Lett ��� Vol ��p �����

��� Potts C N Kovalyov M Y Scheduling with batching� a review WorkingPaper Southampton University

Area �� Problems with operations of equal duration

The no�wait scheduling of n jobs where each job is a chain of unit processingtime operations to be processed alternately on two machines is consideredin ��� The objective is to minimize the mean �ow time An O�n���timealgorithm to produce an optimal schedule is proposed It is also shown thatif zero processing time operations are allowed then the problem is NP �hardin the strong sense

The problem of minimizing the weighted number of late jobs in a two�machinejob�shop with unit time operations is considered in ��� NP �hardness of theproblem is established and a pseudo�polynomial algorithm is proposed whichis based on solving optimally a related problem where a maximal set of earlyjobs is determined

An O�� algorithm for minimizing arbitrary non�decreasing symmetric ob�jective functions for processing n jobs with unit�time operations in an openshop is suggested in ���

In ��� the problem of scheduling a set of unit�time jobs on m uniform ma�chines is studied Some jobs may require a unit of an additional single re�source during their execution The resource is renewable but the total re�source consumption is limited by the same value at each time instant Theobjective is to �nd a feasible schedule minimizing the maximum job com�pletion time It is shown that an approach suggested in the literature tosolve this problem is incorrect An O�m logm� algorithm is presented for

�

the problem with no machine idle times and a linear time algorithm for theproblem with identical machines

The complexity status of a unit time open�shop O j ri� pij � jPUi problem

with release dates and the minimal number of late jobs criterion is investi�gated in ��� It is shown by reduction from ��partition that this problem isunary NP �hard

The complexity status of a three�machine unit�time job�shop J� j pij � jPCi scheduling problem is investigated in ��� It is shown by reduction from

��partition that the problem under consideration is unary NP �hard

A two�machine open�shop problem to minimize the weighted mean �ow timeis considered in ��� Processing times are suggested to be equal to � or � Forthis problem an O�n log n��algorithm is proposed

The proportionate �ow shop problem with controllable processing times isinvestigated in �� In this problem each of the n jobs is processed by mmachines in the same order and all operations of a single job have equal pro�cessing times and may be compressed by the same amount The processingtimes of di�erent jobs may be di�erent The objective is to minimize a sumof the makespan of the schedule and a compression cost function which isnon�decreasing with respect to the amount of compression An O�n log n�algorithm for the following two types of compression cost functions is devel�oped� convex and concave

The single�machine problem of scheduling unit�time jobs with controllablerelease dates is considered in ��� The release dates of the jobs which can becompressed incurring additional costs are given The objective is to mini�mize the makespan of the schedule together with a linear compression costfunction An O�n�� algorithm to construct a trade�o� curve is developed

References

�� Cheng T C E Shakhlevich N V Proportionate Flow Shop with Control�lable Processing Times The Hong Kong Polytechnic University Work�ing Paper No ������ ��� � p �Submitted to Operations Research�

��� Cheng T C E Shakhlevich N V Bicriterion Single Machine Schedulingof the Unit�time Jobs with Controllable Release Dates The Hong KongPolytechnic University Working Paper

��� Kovalyov M Y Shafransky Y M Uniform machine scheduling of unit�time jobs subject to resource constraints Discrete AppliedMathematicsaccepted

��� Kravchenko S A A polynomial algorithm for a two�machine no�waitjob�shop scheduling problem European Journal of Oper Res ���Vol �� ����

�

��� Kravchenko S A Minimizing the number of late jobs for the two�machineunit�time job�shop scheduling problem Discrete Appl Math submit�ted

��� Kravchenko S A On the complexity of minimizing the number of latejobs in unit time open shop Discrete Appl Math submitted

��� Kravchenko S A On the complexity of three�machine job shop mean�ow time problem with unit operations Math Methods of Oper Res submitted

��� Lushchakova I N Kravchenko S A Minimizing Weighted Mean FlowTime in Two Machine Shop Problems with �� Processing Times Eu�ropean Journal of Oper Res accepted

��� Shakhlevich N V Complexity aspects of mixed shop scheduling prob�lems European Journal of Oper Res accepted

Area �� Problems with the �xed number of jobs

A polynomial algorithm for the job�shop scheduling problem with two ma�chines and a �xed number of jobs is developed in ��� The objective is tominimize the sum of weighted completion times It is shown that this prob�lem is solvable in polynomial time

In �� we show that problems J� j n � �� pmtn j Cmax and J� j n � �� pmtn jPCi are show to be binary NP �hard Thus the remaining open question

concerning the complexity status of preemptive job�shop scheduling problemswith �xed numbers of jobs and machines is settled These results are the�rst demonstration that the problem with preemption is binary NP �hardwhile its non�preemptive counterpart is polynomially solvable Furthermorea pseudopolynomial algorithm for problems Jm j n � k� pmtn j Cmax andJm j n � k� pmtn j

PCi is presented which shows that problems J� j n �

�� pmtn j Cmax and J� j n � �� pmtn jPCi cannot be unary NP �hard

For the non�preemptive two�machine job�shop scheduling problem with a�xed number of jobs and any regular objective function polynomial algo�rithms have been developed in ��� This answers previous open questionsabout complexity status of job�shop problems with non�preemptive opera�tions For more special objective functions structural properties of optimalschedules were derived These properties are useful for the development ofbranch�and�bound methods

The mixed�shop problems with a �xed number of jobs are considered in�� � �� In the mixed shop some jobs are processed with �xed routes likein the job shop while the routes of the others are not �xed as in the openshop In ��� results developed for mixed�shop problems with an unlimited

�

number of operations per job are presented In �� �� results on computa�tional complexity of mixed�shop problems are surveyed The main attentionis devoted to establishing the boundary between polynomially solvable andNP �hard problems

A complexity analysis for the mixed�shop problems with a �xed number ofjobs is done in ��� Problem X�jnJ � �� pmtnjCmax is considered and anO�n� r�� algorithm is proposed It is a preemptive two�machine mixed�shopproblem with nJ � � jobs processed like in the job shop and an unlimitednumber nO of jobs processed like in the open shop Here n � nJ � nO isthe overall number of jobs and r is the maximal number of operations forjobs which are processed like in the job shop Binary NP �hardness of prob�lem X�jnJ � �� nO � � pmtnjCmax �preemptive three�machine mixed�shopproblem with nJ � � and nO � � is established and a pseudopolynomialalgorithm is proposed A number of algorithms for non�preemptive mixed�shop problems are developed on the basis of algorithms for non�preemptive�pure shop problems

References

�� Brucker P Kravchenko S A Sotskov Yu N Preemptive job�shopscheduling problems with a �xed number of jobs Mathematical Modelsof Operations Research �ZOR� accepted

��� Brucker P Kravchenko S A Sotskov Yu N On the complexity of twomachine job�shop scheduling with regular objective functions OR Spec�trum ��� Vol � p ���

��� Kravchenko S A Sotskov Y N Complexity of the two machine job�shopscheduling problem with a �xed number of jobs Central European J Oper Res Economics submitted

��� Shakhlevich N V Complexity of the Mixed Shop Scheduling Problems�a Survey XV EURO Summer Institute September�October ��� Italy

��� Shakhlevich N V Sotskov Yu N Werner F Shop�scheduling problemswith �xed and non�xed orders of the jobs Annals of Oper Res sub�mitted

��� Shakhlevich N V Sotskov Y N Werner F Shop�scheduling problemswith �xed and non��xed orders of the jobs Preprint N � �Otto�von�Guericke�Universit�at Magdeburg ��� �� p �

��� Shakhlevich N V Sotskov Y N Werner F Complexity of Mixed ShopScheduling Problems� a Survey European Journal of Operational Re�search submitted

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Area �� Generalized shop scheduling problems

In �� a generalized shop scheduling problem is considered where arbi�trary precedence constraints among the operations are given �general shop�sequence�dependent changeover times between the processing of operationsare taken into consideration multiprocessor requirements of operations canoccur and in each technological stage of a job several identical machines ofthe corresponding type may be available �hybrid or �exible shop� Howeverthe number of used machines of each type is not a priori �xed As an ob�jective function an arbitrarily given monotonically non�decreasing functionin the completion times of the operations and the numbers of used identicalmachines of each type is considered Based on the mixed multigraph modeltwo heuristic decomposition procedures and several algorithms for construct�ing and improving a schedule are proposed The developed algorithms havebeen tested on a set of problem instances with up to �� jobs and up to �machine types

A scheduling problem in which each job has to be prepared before it can beprocessed is considered in ��� The preparation is performed by a batchingmachine� it can prepare at most c jobs in each run which takes an amount oftime that is independent of the number and identity of the jobs under prepa�ration A strong Lagrangian lower bound is presented by using a new conceptof positional completion times This bound can be computed in O�n log n�time only where n is the number of jobs Two classes of O�n log n� heuristicsthat transform an optimal schedule for the Lagrangian dual problem into afeasible schedule are presented Any heuristic in one class has a performanceguarantee of ��� It is shown that even the most naive heuristic in this classhas a compelling empirical performance

The problem of scheduling jobs on parallel machines with set�up times isconsidered in ��� The set up has to be performed by a single server Theobjective is to minimize the schedule length �makespan� as well as the forcedidle time A pseudopolynomial algorithm for the case of two machines whenall set�up times are equal to one is presented It is shown that the moregeneral problem with an arbitrary number of machines is unary NP �hard andsome list scheduling heuristics are analyzed for this problem The problemof minimizing the forced idle time is known to be unary NP �hard for thecase of two machines and arbitrary set�up and processing times Unary NP �hardness of this problem even for the case of constant set�up times is provedand some polynomially solvable cases are given

Scheduling problems to minimize the makespan provided that the set�upphase of each operation needs to be attended by a single server commonfor all jobs are considered in ��� For the processing system consisting oftwo parallel dedicated machines it is proved that the problem of �nding anoptimal schedule is NP �hard in the strong sense even if all set�up times areequal or if all processing times are equal For the case of m parallel dedicatedmachines a simple greedy algorithm is shown to create a schedule with the

�

makespan that is at most twice the optimumvalue For the two�machine casean improved heuristic guarantees a tight worst�case ratio of ��� Severalpolynomially solvable cases of the latter problem are described The two�machine �ow shop and the open�shop problems with a single server are alsoshown to be NP �hard in the strong sense However the two�machine �ow�shop no�wait problem with a single server reduces to the Gilmore�Gomorytraveling salesman problem and is therefore solvable in polynomial time

The problem of processing n jobs in a two�machine non�preemptive openshop to minimize the makespan is considered in ��� One of the machines isassumed to be non�bottleneck It is shown that unlike its �ow shop coun�terpart the problem is NP �hard in the ordinary sense On the other handthe problem is shown to be solvable by a dynamic programming algorithmthat requires pseudopolynomial time The latter algorithm can be convertedinto a fully polynomial approximation scheme that runs in O�n���� time AnO�n log n� approximation algorithm is also designed which �nds a schedulewith makespan at most ��� times the optimal value and this bound is tight

Two models of two�stage processing with no�wait in the process are studiedin ��� The �rst model is the two�machine �ow shop and the other is theassembly model For both models the problem of minimizing the makespanprovided that the set�up and removal times are separated from the processingtimes is considered Each of these scheduling problems is reduced to thetraveling salesman problem �TSP� It is shown that in general the assemblyproblem isNP �hard in the strong sense On the other hand the two�machine�ow shop problem reduces to the Gilmore�Gomory TSP and is solvable inpolynomial time The same holds for the assembly problem under somereasonable assumptions

A problem of sequencing n jobs in a two�machine re�entrant shop with theobjective of minimizing the maximum completion time is considered in �� The shop consists of two machines M� and M� and each job has the pro�cessing route �M�� M��M�� An O�n log n� time heuristic is presented whichgenerates a schedule with length at most ��� times that of an optimal sched�ule thereby improving the best previously available worst�case performanceratio of ���

Two �minimum NP �hard job�shop scheduling problems to minimize themakespan are investigated in ��� In one of the problems every job has to beprocessed on at most two out of three available machines In the other prob�lem there are two machines and a job may visit one of the machines twice For each problem a class of heuristic schedules is de�ned in which certainsubsets of operations are kept as blocks on the corresponding machines It isshown that for each problem the value of the makespan of the best schedulein that class cannot be substantially less than ��� times the optimal value Algorithms that guarantee a worst�case ratio of ��� are presented

The two�stage m�machine scheduling models with a bottleneck machine arestudied in �� � �� Here the processing of each of n jobs involves only two

�

operations one of which is to be processed on a given machine while theother operation must be processed on the bottleneck machine common for alljobs In the case of the open shop it is shown that the problem of minimizingthe makespan is solvable in O�n� time if m � �� see ��� In the case of anarbitrary m a heuristic algorithm is developed that delivers a worst�caseperformance ratio of ��� see ��� For the job�shop version of the problem aheuristic algorithm described in ��� guarantees a worst�case ratio of ���

The problem of minimizing the schedule length of a two�machine open shopin which a job can not only be assigned any of the two possible routes butalso the processing times depend on the chosen route is considered in ��� This problem is known to be NP �hard A simple approximation algorithmthat guarantees a worst�case performance ratio of � is proposed Some mod�i�cations to this algorithm that improve its performance and guarantee aworst�case performance ratio of ��� are given

The open�shop scheduling problem to minimize the makespan provided thatone of the machines has to process the jobs according to a given sequenceis studied in ��� It is shown that in the preemptive case the problem ispolynomially solvable for an arbitrary number of machines If preemption isnot allowed the problem is NP �hard in the strong sense if the number ofmachines is variable and is NP �hard in the ordinary sense in the case of twomachines For the latter case a heuristic algorithm is suggested that runsin linear time and produces a schedule with a makespan that is at most ���times the optimal value It is also shown that the two�machine problem inthe non�preemptive case is solvable in pseudopolynomial time by a dynamicprogramming algorithm and that the algorithm can be converted into a fullypolynomial approximation scheme

An extension of the greedy approach to �nding an approximate solution toan open�shop problem to minimize the makespan is considered in ��� Ingeneral the algorithm reduces either a relative or an absolute error of thetraditional greedy approach and in the case of three machines guarantees aworst�case ratio of ���

The two�machine open�shop problem to minimize the makespan providedthat the jobs are transported from one stage to the other is studied in ���and ��� The �rst of these papers develops a linear time algorithm providedthat the transportation times are small and presents a heuristic for job�independent transportation times The case of arbitrary times is handled in��� where an algorithm with a worst�case ratio of ��� is given

In ���� the problem of scheduling n jobs in a pallet�constrained �ow shop withthe objective to minimize the makespan is considered In such a �ow shopenvironment each job needs a pallet the entire time from the start of its �rstoperation until the completion of the last operation and the number of pal�lets in the shop at any given time is limited by an integerK with � K � n Primarily the two�machine �owshop is considered and speci�cally the impactof the number of pallets on the makespan is studied While it is an NP �hard

problem to �nd the minimum number of pallets subject to an upper boundon the makespan a worst�case bound on the minimumK that guarantees theleast possible makespan is proved Furthermore the empirical performanceof Johnson�s algorithm is investigated which solves the problem to optimal�ity if K � n and Gilmory�Gomory�s algorithm which solves the problem tooptimality if K � � when they are both straightforwardly adapted to dealwith the situation where there are only K pallets available at any time andwhere � � K � n Computational experiments with randomly generatedinstances reveal that for Johnson�s algorithm which produces the least pos�sible makespan the required number of pallets grows quite rapidly with thenumber of jobs In contrast Gilmore�Gomory�s algorithm which may fail toproduce the least possible makespan consistently requires only about fourpallets to produce schedules with a makespan very close to optimal

In ��� we study the special case of an m machine �ow shop problem in whichthe processing time of each operation of job j is equal to pj is studied� thisvariant of the �ow shop problem is known as the proportionate �ow shopproblem It is shown that for any number of machines and for any regularperformance criterion one can restrict the search for an optimal schedule topermutation schedules Moreover the problem of minimizing total weightedcompletion time is solvable in O�n�� time

In ��� a generalization of the permutation �ow shop problem of schedulingn jobs on m machine centers is considered which is more adequate to someproduction scheduling problems e g in the copper industry Each centerconsists of parallel identical machines each job consists of ordered opera�tions that have to be processed on machines from di�erent machine centersthe sequence in which the jobs pass through the machine centers is identicallygiven for all jobs The problem is to �nd an optimal sequence �schedule� inwhich n jobs pass through each machine center this sequence having to bethe same for each center In addition to the hybrid �ow�shop problem thefollowing generalizations are also considered which make the problem moreadequate to practical use The processing times of the operations on somemachines may vary between a minimum and a maximum value dependingon the use of a continuously�divisible resource �this makes the problem morerelevant for scheduling a copper electrore�ning process in the copper indus�try� A non�regular criterion is based on due dates which are not a priorigiven but can be �xed by a decision�maker A due date assignment cost isincluded into the objective function Some properties of non�regular criteriafor such a problem have been studied

A job�shop scheduling problem with the objective of minimizing the sum ofjob completion times under uncertain numerical data is modeled in terms ofa mixed graph in ��� In contrast to well�studied deterministic settings itis assumed that only the structural input data �i e precedence and capac�ity constraints� are �xed while for the operation processing times only theirlower and upper bounds are known before scheduling and the probabilitydistribution functions of the random processing times are unknown The

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structural input data are uniquely de�ned by the technological routes of thejobs e g for a �ow or open shop �xing the structural input data simplymeans �xing the numbers of jobs and machines A mixed graph formulationof such a problem and an approach for dealing with such !strict uncertainty�based on a stability analysis of an optimal schedule are developed Domi�nance and strong dominance relations on the set of schedules are introducedand investigated These relations are used for characterizations of a solutionand a minimal solution of the scheduling problem with uncertain processingtimes The formula for calculating the stability radius of an optimal sched�ule is given The obtained results are used in a procedure for constructingpotentially optimal schedules for all perturbations of the processing timeswithin the given lower and upper bounds To exclude redundant schedules astability analysis was used on the basis of pairwise comparison of schedules

References

�� Drobouchevitch I G Strusevich V A A heuristic algorithm for the two�machine re�entrant shop scheduling Annals of Oper Res accepted

��� Drobouchevitch I G Strusevich V A Heuristics for short route job�shopscheduling problems Mathematical Methods of Oper Res accepted

��� Drobouchevitch I G Strusevich V A A polynomial algorithm for thethree�machine open shop with a bottleneck machine Annals of Oper Res accepted

��� Drobouchevitch I G Strusevich V A Two�stage open shop schedulingproblem with a bottleneck machine INFORMS J on Computing sub�mitted

��� Drobouchevitch I G Strusevich V A Heuristics for the two�stage jobshop scheduling problem with a bottleneck machine European J Oper Res submitted

��� Glass C A Shafransky Y M Strusevich V A Scheduling for paralleldedicated machines with a single server SIAM Journal on Computingsubmitted

��� Gupta J N D Kr�uger K Lau� V Sotskov Yu N Werner F Heuristicsfor hybrid �ow shops with controllable processing times and assignabledue dates IEEE Transactions submitted

��� Gupta J N D Strusevich V A Zwaneveld C M Two�stage no�waitscheduling models with setup and removal times separated Comput�ers and Oper Res ��� Vol �� p ������

�

��� Hoogeveen J A van de Velde S L Scheduling by positional completiontimes� analysis of a two�stage �ow shop problem with a batching ma�chine Mathematical Programming accepted

��� Kravchenko S A Werner F Parallel machine scheduling problems witha single server Math Comput Modelling ��� Vol �� N � p �

�� Kr�uger K Sotskov Y N Werner F Heuristics for generalized shopscheduling problems based on decomposition International Journal ofProduction Res accepted

��� Lai T �C Sotskov Yu N Sotskova N Yu Werner F Mean �ow timeminimization with uncertain processing times Naval Research Logisticssubmitted

��� Rebaine D StrusevichV A Open shop scheduling problems with specialtransportation times J Oper Res Society submitted

��� Shafransky Y M Strusevich V A The open shop scheduling problemwith a given sequence of jobs on one machine Naval Res Logisticsaccepted

��� Shakhlevich N Hoogeveen H Pinedo M Minimizing total weightedcompletion time in a proportionate �ow shop Journal of Schedulingsubmitted

��� Strusevich V A A greedy open shop heuristic with job priorities Annalsof Oper Res accepted

��� Strusevich V A A heuristic for the two�machine open shop schedulingproblem with special transportation times Discr Appl Math submit�ted

��� Strusevich V A Hall L A An open shop scheduling problem with anon�bottleneck machine Oper Res Lett ��� Vol � p ��

��� Strusevich V A van de Waart A J A Dekker R Heuristic algorithmsfor the two�machine open shop scheduling problem with route�dependentprocessing times Journal of Heuristics accepted

���� Wang M W Sethi S P Sriskandarajah C van de Velde S HoogeveenH Minimizing makespan in a pallet�constrained �owshop Journal ofScheduling submitted

Area �� Local search heuristics

In ��� a new binary encoding scheme for a single�machine scheduling problemto minimize total weighted tardiness has been introduced and tested in con�nection with multi�start descent simulated annealing threshold accepting

�

tabu search and genetic algorithms on large sets of test problems The com�putational results show that the binary representation is more robust thanthe usual �natural permutation representation

A new technique called dynasearch is proposed in ��� which uses dynamicprogramming to search for the best move in a rather large neighborhood The application of local search to problems which are a combination of asequencing problem and another optimization problem are considered in �� For one problem mentioned in �� an e�cient procedure to calculate the bestneighbor of a given solution for the adjacent pair interchange neighborhood isgiven in ��� This procedure reduces the complexity of �nding the best neigh�bor from quadratic to linear time Based on these results a technique similarto dynasearch is used to calculate the best neighbor for a new neighborhoodof exponential size in quadratic time

In ��� a tabu�search algorithm is presented and computational results arereported for the multi�mode job�shop problem The multi�mode job�shopproblem is a version of the multi�processor�task job�shop problem in whicha set of machines must be assigned to each operation The operation usedall the assigned machines simultaneously during its processing time whichdepends on the assigned machines

References

�� Brucker P Hurink J Complex Sequencing problems and local searchheuristics in Metaheuristics� Theory and Applications Edited by I H Osman and J P Kelly Kluwer Academic Publishers Boston ���p ����

��� Brucker P Neyer J Tabu�search for the multi�mode job�shop problemOR Spektrum ��� Vol �� p ����

��� Crauwels H A J Potts C N VanWassenhove L N Local Search Heuris�tics for the Single Machine Total Weighted Tardiness Scheduling Prob�lem INFORMS Journal on Computing ��� Vol � to appear

��� Hurink J E�cient Calculation of a Best Neighbor for a One MachineBatching Problem Osnabr�ucker Schriften zur Mathematik Reihe PHeft �� ���

��� Potts C S van de Velde S Dynasearch�Iterative Improvement by Dy�namic Programming� Part I The Traveling Salesman Problem ���Working Paper

�

Area � Singlemachine preemptive scheduling prob

lems with deadlines� release and due dates

A single�machine preemptive scheduling problem to minimize the weightednumber of late jobs is considered in ��� Given are n jobs and for each jobwe have a release date a processing time and a due date It is assumed thatcertain speci�ed jobs have to be completed on time The due dates for thesejobs are considered as deadlines while for the other jobs due dates may beviolated For the case of similarly ordered release and due dates �when thereis no advantage to preemption� as well as for the case of oppositely orderedrelease and due dates the initial problem is transformed into a reduced prob�lem where all jobs with deadlines are eliminated O�n log n� algorithms areproposed for some special cases of the considered problems for example whenprocessing times and weights are oppositely ordered

The single�machine due date assignment and scheduling problems with njobs in which the due dates are to be obtained from the processing times byadding a positive slack q are studied in ��� A schedule is feasible if thereare no tardy jobs and the job sequence respects given precedence constraints The value of q is chosen so as to minimize a function ��F� q� which is non�decreasing in each of its arguments where F is a certain non�decreasingearliness penalty function Once q is chosen or �xed the correspondingscheduling problem is to �nd a feasible schedule with the minimum F �valueof function F In the case of arbitrary precedence constraints the problemsunder consideration are shown to be NP �hard in the strong sense even whenF is total earliness If the precedence constraints are de�ned by a series�parallel graph both scheduling and due date assignment problems are provedto be solvable in O�n� log n� time provided that F is either the sum of linearfunctions or the sum of exponential functions The running time of thealgorithms can be reduced to O�n log n� if the jobs are independent

The single�machine problem of scheduling n independent jobs subject totarget starting times is discussed in ��� Target starting times are essentiallyrelease times that may be violated at a certain cost The objective is tominimize an objective function that is composed of total completion timeand maximum promptness which measures the observance of these targetstarting times It is shown that in the case of a linear objective functionthe problem is solvable in O�n�� time if preemption is allowed or if the totalcompletion time outweights maximum promptness

Column generation has proved to be an e�ective technique for solving thelinear programming relaxation of huge set covering or set partitioning prob�lems and column generation approaches have led to state�of�the�art so�calledbranch�and�price algorithms for various archetypical combinatorial optimiza�tion problems Usually if Lagrangean relaxation is embedded at all in acolumn generation approach then the Lagrangean bound serves only as atool to fathom nodes of the branch�and�price tree In �� it is shown that theLagrangean bound can be exploited in more sophisticated and e�ective ways

�

for two purposes� to speed up convergence of the column generation algo�rithm and to speed up the pricing algorithm Our vehicle to demonstrate thee�ectiveness of teaming up column generation with Lagrangean relaxation isan archetypical single�machine common due date scheduling problem Ourcomprehensive computational study shows that the combined algorithm is byfar superior to two existing purely column generation algorithms� it solves in�stances with up to �� jobs to optimality while the purely column generationalgorithm can solve instances with up to only �� jobs

In ��� and ��� fully polynomial approximation schemes are derived for theproblem of scheduling n jobs on a single machine to minimize total weightedearliness and tardiness and for the problem of scheduling n deteriorating jobson a single machine to minimize makespan respectively A new technique isused to develop the schemes The main feature of this technique is that itrecursively computes lower and upper bounds on the value of partial optimalsolutions Therefore the scheme does not require any prior knowledge oflower and upper bounds on the value of a complete optimal solution

References

�� van den Akker J M Hoogeveen J A van de Velde S L Combining col�umn generation and Lagrangean relaxation� an application to a single�machine common due date scheduling problem INFORMS Journal onComputing submitted

��� Gordon V Potapneva E Werner F Single machine scheduling withdeadlines release and due dates Optimization ��� Vol �� p ������

��� Gordon V S Strusevich V A Earliness penalties on a single machinesubject to precedence constraints� SLK due date assignment Comp Oper Res submitted

��� Hoogeveen J A van de Velde S L Scheduling with target start times Oper Res Lett submitted

��� Kovalyov M Y Kubiak W A fully polynomial approximation schemefor weighted earliness�tardiness problem Operations Research to ap�pear

��� Kovalyov M Y Kubiak W ����� A fully polynomial approximationscheme for minimizingmakespan of deteriorating jobs Journal of Heuris�tics ��� Vol � �������

�

Area �� Stability analysis

The usual assumption that the processing times of the operations are knownin advance is the strictest one in deterministic scheduling theory and it essen�tially restricts its practical aspects A stability analysis of an optimal sched�ule may help to extend the signi�cance of scheduling theory for some produc�tion scheduling problems The terms �stability� �sensitivity� or �postoptimalanalysis� are generally used for the phase of an algorithm at which a solution�or solutions� of an optimization problem has already been found and ad�ditional calculations are performed in order to investigate how this solutiondepends on the problem data The stability radius of an optimal schedulefor a general shop scheduling problem denotes the largest quantity of inde�pendent variations of the processing times of the operations such that thisschedule remains optimal

The calculation of the stability radius of an optimal or an approximate sched�ule for job shop scheduling problems is considered in ��� Computationalresults on the calculation of the stability radius for randomly generated job�shop scheduling problems are presented The extreme values of the stabilityradius are considered in more detail

The in�uence of errors and possible changes of the processing times onthe property of a schedule to be optimal for problems J jprecj

PCi and

J jprecjCmax is investigated in ��� A branch�and�bound method for the calcu�lation of the stability radius of optimal schedules is used Extensive numericalexperiments with randomly generated job�shop scheduling problems are per�formed and discussed Due to the developed software we have the possibilityto compare the values of the stability radii the numbers of optimal schedulesand some other �numbers� for two often used criteria The main question ishow large the stability radius is on average for randomly generated job�shopproblems

Formulas for the calculation of the stability radius when the objective is tominimizemean or maximum�ow time for a general shop scheduling problemare given in ��� The extreme values of the stability radius are of particularimportance and these are considered in more detail Computational resultson the calculation of the stability radius for randomly generated job�shopscheduling problems are discussed

The general shop problem with mean �ow time objective function is con�sidered in �� Necessary and su�cient conditions for the stability radius ofoptimal schedule to be equal to zero or to in�nity are obtained The formulafor calculating the exact value of the stability radius is proposed

In ��� a scheduling problem is considered with the objective of minimizingthe makespan under uncertain numerical input data The processing timeof an operation the release time and due date of a job are usually eitherassumed to be �xed before the scheduling in a deterministic model or totake some known probability distribution functions in a stochastic model

�

This paper deals with the scheduling problem in which the structural inputdata �i e precedence and capacity constraints� are �xed before the stage ofscheduling while all that is known before scheduling about the processingtime of an operation the job release time and due date are their upperand lower bounds After improving the mixed graph model an approach isdeveloped for dealing with the scheduling problem under uncertain numericaldata based on a stability analysis of an optimal makespan schedule Inparticular the candidate set of the critical paths in a circuit�free digraph isinvestigated a minimal set of the optimal schedules is characterized and anoptimal and a heuristic algorithm are developed Computational results forrandomly generated as well as well�known test problems are discussed

Paper ��� deals with the general shop scheduling problem with the objec�tive of minimizing the makespan under uncertain scheduling environments The processing time of an operation is usually assumed to take a knownprobability distribution function when dealing with uncertain scheduling en�vironments The scheduling environments are supposed to be so uncertainthat all information available about the processing time of an operation is anupper and lower bound An approach to deal with such a situation based onan improved stability analysis of an optimal makespan schedule is presentedand this approach is demonstrated on an illustrative example of the job�shopscheduling problem

References

�� Br�asel H Sotskov Y N Werner F Stability of a schedule minimizingmean �ow time Mathematical and Computer Modelling ��� Vol ��p �����

��� Lai T �C Sotskov Yu N Sequencing with uncertain numerical data formakespan minimization J Oper Res Soc submitted

��� Lai T �C Sotskov Yu N Sotskova N Yu Werner F Optimal makespanscheduling with given bounds of processing times Mathematical andComputer Modelling ��� Vol �� No � p �����

��� Sotskov Yu N Sotskova N Y Werner F Stability of an optimal sched�ule in a job shop OMEGA ��� Vol �� No � p ������

��� Sotskov Yu N Tanaev V S Werner F Stability radius of an optimalschedule� a survey and recent developments Industrial Application ofCombinatorial Optimization Netherlands �Kluwer Academic Publish�ers accepted

��� Sotskov Yu N Wagelmans A P M Werner F On the calculation of thestability radius of an optimal or an approximate schedule Annals ofOper Res accepted

�

List of published and accepted papers

Azharonok E Gordon V Werner F Single machine preemptive schedulingwith special cost functions Optimization ��� Vil �� p ������

Br�asel H Shakhlevich N V On the solution region for certain schedulingproblems with preemption Annals of Oper Res accepted

Br�asel H Sotskov Y N Werner F Stability of a schedule minimizingmean�ow time Mathematical and Computer Modelling ��� Vol �� p �����

Brucker P Gladky A Hoogeveen H Kovalyov M Y Potts C TautenhahnT van de Velde S Scheduling a batching machine Journal of Scheduling��� Vol p ����

Brucker P Jurisch B Kr�amer A Complexity of scheduling problems withmulti�purpose machines Annals of Oper Res ��� Vol �� p �����

Brucker P Kovalyov M Y Single machine batch scheduling to minimizethe weighted number of late jobs Mathematical Methods of Oper Res ��� Vol �� p ��

Brucker P Kovalyov M Y Shafransky Y M Werner F Parallel machinedeadline batch scheduling Annals of Oper Res accepted

Brucker P Kravchenko S A Sotskov Yu N On the complexity of two ma�chine job�shop scheduling with regular objective functions OR Spektrum��� Vol � p ���

Brucker P Thiele O A branch " bound method for the general shop prob�lem with sequence dependent setup�times OR�Spektrum ��� Vol � p ����

Chen B Potts C N Woeginger G J A review of machine scheduling� com�plexity algroithms and approximability in� D �Z Du and P M Pardalos�eds � Handbooks of Combinatorial Optimization Kluwer ��� to appear

Cheng T C E Gordon V S Kovalyov M Y Single machine scheduling withbatch deliveries European Journal of Oper Res ��� Vol �� p �������

Cheng T C E Janiak A Kovalyov M Y Bicriterion single machine schedul�ing with resource dependent processing times SIAM Journal on Optimiza�tion accepted

Cheng T C E Kovalyov M Y Complexity of parallel machine schedulingwith processing�plus�wait due dates to minimize maximum absolute latenessEuropean Journal of Operations Research accepted

��

Cheng T C E Kovalyov M Y An exact algorithm for batching and schedul�ing two part types in a mixed shop� a technical note International Journalof production Economics accepted

Cheng T C E Kovalyov M Y Lin B M T Single machine scheduling tominimize batch delivery and job earliness penalties SIAM Journal on Opti�mization ��� Vol � p �������

Chen B Potts C N Strusevich V A Approximation algorithms for two�machine �ow shop scheduling with batch setup times Mathematical Pro�gramming accepted

Crauwels H A J Hariri A M A Potts C N Van Wassenhove L N Branchand bound algorithms for single machine scheduling with batch set�up timesto minimize total weighted completion time Annals of Operations Researchaccepted

Crauwels H A J Potts C N Van Wassenhove L N Local search heuristicsfor single machine scheduling with batching to minimize the number of latejobs European Journal of Oper Res ��� Vol �� p ������

Crauwels H A J Potts C N Van Wassenhove L N Local search heuris�tics for single machine scheduling with batch set�up times to minimize totalweighted completion time Annals of Oper Res ��� Vol �� p ������

Crauwels H A J Potts C N Van Wassenhove L N Local search heuristicsfor single machine total weighted tardiness scheduling problem INFORMSJournal on Computing ��� Vol � to appear

Drobouchevitch I G Strusevich V A A heuristic algorithm for the two�machine re�entrant shop scheduling Annals of Oper Res accepted

Drobouchevitch I G Strusevich V A Heuristics for short route job shopscheduling problems Mathematical Methods of Oper Res accepted

Drobouchevitch I G Strusevich V A A polynomial algorithm for the three�machine open shop with a bottleneck machine Annals of Oper Res ac�cepted

Gordon V Potapneva E Werner F Single machine scheduling with dead�lines release and due dates Optimization ��� Vol �� p ������

Gupta J N D Strusevich V A Zwaneveld C M Two�stage no�wait schedul�ing models with setup and removal times separated Computers and Oper Res ��� Vol �� p ������

Hoogeveen J A van de Velde S L Scheduling by positional completion times�

�

analysis of a two�stage �ow shop problem with a batching machine Mathe�matical Programming accepted

Kovalyov M Y Kubiak W A fully polynomial approximation scheme forminimizing makespan of deteriorating jobs Journal of Heuristics ��� Vol � p �������

Kovalyov M Y Kubiak W A fully polynomial approximation scheme forweighted earliness�tardiness problem Operations Research accepted

Kovalyov M Y Shafransky Y M Uniform machine scheduling of unit�timejobs subject to resource constraints Discrete Applied Mathematics ac�cepted

Kovalyov M Y Shafransky Y M Batch scheduling with deadlines on parallelmachines� An NP �hard case Inform Proc Lett ��� Vol �� p �����

Kovalyov M Y Werner F A polynomial approximation scheme for problemF�jrjjCmax Oper Res Lett ��� Vol �� N� p �����

Kravchenko S A A polynomial algorithm for a two�machine no�wait job�shop scheduling problem European Journal of Oper Res ��� Vol ������

Kravchenko S A Werner F Parallel machine scheduling problems with asingle server Math Comput Modelling ��� Vol �� N � p �

Kr�uger K Shakhlevich N Sotskov Yu N Werner F A decomposition algo�rithm for scheduling problems on mixed graphs Journal of the Oper Res Soc ��� Vol �� p ������

Kr�uger K Sotskov Y N Werner F Heuristics for generalized shop schedul�ing problems based on decomposition International Journal of ProductionRes accepted

Lai T �C Sotskov Yu N Sotskova N Yu Werner F Optimal makespanscheduling with given bounds of processing times Mathematical and Com�puter Modelling ��� Vol �� No � p �����

Lushchakova I N Kravchenko S A MinimizingWeightedMean Flow Time inTwo Machine Shop Problems with �� Processing Times European Journalof Oper Res accepted

Shafransky Y M Strusevich V A The open shop scheduling problem with agiven sequence of jobs on one machine Naval Res Logistics accepted

Shakhlevich N V Complexity aspects of mixed shop scheduling problems

��

European Journal of Oper Res accepted

Shakhlevich N V Sotskov Yu N Werner F Adaptive scheduling algorithmbased on the mixed graph model IEE Proceedings � Control Theory Appl ��� Vol �� N p ���

Sotskov Yu N Software for production scheduling based on the mixed �mul�ti�graph approach Computing and Control Engineering Journal ��� Vol � N� p �������

Sotskov Yu N Mixed multigraph approach to scheduling jobs on machinesof di�erent types Optimization ��� Vol �� p �������

Sotskov Yu N Tanaev V S Werner F Stability radius of an optimal sched�ule� a survey and recent developments Industrial Application of Combina�torial Optimization Netherlands �Kluwer Academic Publishers accepted

Sotskov Yu N Sotskova N Y Werner F Stability of an optimal schedule ina job shop OMEGA ��� Vol �� No � p ������

Sotskov Yu N Tautenhahn T Werner F On the application of insertiontechniques for job shop problems with setup times Oper Res accepted

Sotskov Yu N Tautenhahn T Werner F Heuristics for permutation �owshop scheduling with bath setup times OR Spektrum ��� Vol � p �����

Sotskov Yu N Wagelmans A P M Werner F On the calculation of thestability radius of an optimal or an approximate schedule Annals of Oper Res accepted

Strusevich V A A greedy open shop heuristic with job priorities Annals ofOper Res accepted

Strusevich V A Hall L A An open shop scheduling problem with a non�bottleneck machine Oper Res Lett ��� Vol � p ��

Strusevich V A van de Waart A J A Dekker R Heuristic algorithms for thetwo�machine open shop scheduling problem with route�dependent processingtimes Journal of Heuristics accepted

Workshops and conferences

The �Third Workshop on Models and Algorithms for Planning and Schedul�ing Problems was organized with the participation of C Glass C Potts andV Strusevich in Cambridge United Kingdom April �� ��� At this inter�national workshop scientists from � countries discussed scheduling problems

��

� members of the di�erent groups within the INTAS�Project participatedin the workshop and presented new project results

On April � ��� a business meeting for the INTAS�Project was organized Members of the project groups discussed research projects and organizedvisits for NIS participants

For the �th European Conference on Operational Research which took placein Brussels July ��� ��� three INTAS Sessions were organized by the co�ordinator

Summary

Constraint Scheduling� Mathematical models� algorithms� and ap�

plications

Reference number� INTAS������� ext

Project co�ordinator� Peter Brucker

Period� June ��� to May � ���

Many scheduling problems that arise in production planning systems containparameters and conditions not present in classical scheduling environments To solve such problems it is necessary to construct speci�c methods andalgorithms Also a technique for their complexity analysis is to be developed Finding relevant methods and techniques is the objective of the project

For a number of generalized shop scheduling problems where such parame�ters as set�up times parallel machines on di�erent processing stages di�er�ent types of jobs their complexity status has been established heuristic andapproximate algorithms with guaranteed ratio performance have been sug�gested For some special cases polynomial time algorithms for solving themto optimality have been given Also the stability of optimal and approximatesolutions for shop scheduling problems have been investigated

The complexity of batch scheduling problems was studied and approximationalgorithms for their solution developed

In many situations problems with equal duration of operations or jobs arise In another typical situation we have a �xed number of jobs in the processingsystem For a number of such problems their complexity status has beenestablished and exact and approximate polynomial time algorithms have beenproposed

For most of the complex scheduling problems considered it is impossible to�nd exact solutions in reasonable time Therefore local search heuristics weredeveloped and tested for some of the problems

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